The filtered martingale problem
Thomas G. Kurtz ∗
Departments of Mathematics and Statistics
University of Wisconsin - Madison
480 Lincoln Drive
Madison, WI 53706-1388
USA
[email protected]
http://www.math.wisc.edu/∼kurtz/
Giovanna Nappo †
Dipartimento di Matematica “Guido Castelnuovo”
Università degli Studi LA SAPIENZA
piazzale A. Moro 2
00185 Roma
ITALY
[email protected]
http://www.mat.uniroma1.it/people/nappo/nappo.html
May 24, 2010
Abstract
Let X be a Markov process characterized as the solution of a martingale problem
with generator A, and let Y be a related observation process. The conditional distribution πt of X(t) given observations of Y up to time t satisfies certain martingale
properties, and it is shown that any probability-measure-valued process with the appropriate martingale properties can be interpreted as the conditional distribution of
X for some observation process. In particular, if Y (t) = γ(X(t)) for some measurable mapping γ, the conditional distribution of X(t) given observations of Y up to
time t is characterized as the solution of a filtered martingale problem. Uniqueness for
the original martingale problem implies uniqueness for the filtered martingale problem
which in turn implies the Markov property for the conditional distribution considered
as a probability-measure-valued process. Other applications include a Markov mapping
theorem and uniqueness for filtering equations.
MSC 2000 subject classifications: 60J25, 93E11, 60G35, 60J35, 60G44
Keywords: partial observation, conditional distribution, martingale problems, filtering, filtering equation, Markov process, Markov mapping
∗
Research supported in part by NSF grants DMS 05-03983 and 08-05793
Research supported in part by Progetto Università - 2008 - prot.C26A08H8YM, Interazione e dipendenza
nei modelli stocastici
†
1
1
Introduction
The notion of a filtered martingale problem was introduced in Kurtz and Ocone (1988) and
extended to a more general setting in Kurtz (1998). The basic idea is that the conditional
distribution of the state of a Markov process given the information from related observations
satisfies a kind of martingale problem. The fundamental results give conditions under which
every solution of the filtered martingale problem arises from a solution of the original martingale problem, and hence, uniqueness for the original martingale problem implies uniqueness
for the filtered martingale problem. These results have a variety of consequences, most notably, the uniqueness for filtering equations and general results on Markov mappings, that
is, conditions under which a transformation of a Markov process is still Markov.
The current paper is concerned with extension of these ideas to further settings. The filtering literature contains a number of results (for example, Bhatt, Budhiraja, and Karandikar
(2000); Budhiraja (2003); Kunita (1971)) showing that the conditional distribution for the
classical filtering problem is itself a Markov process. In order to address this question for
general solutions of filtered martingale problems, we need to generalize the earlier definition
to include information available at time zero. We are then able to show the Markov property
for the conditional distributions for a large class of partially observed Markov processes.
We also extend the earlier results to local martingale problems which in turn allows us
to generalize previous uniqueness results for filtering equations. The basic results can also
be extended to constrained martingale problems, that is, martingale problems for processes
in which the behavior of the process on the boundary of the state space is determined by a
second operator (see Anderson (1976); Kurtz (1990, 1991); Kurtz and Stockbridge (2001);
Stroock and Varadhan (1971)). Reflecting diffusion processes provide one example.
We also relax some technical conditions present in the earlier work.
Throughout this paper, all filtrations are assumed complete, all processes are assumed
to be progressively measurable, {FtY } denotes the completion of the filtration generated by
theR observed process Y , and assuming Y takes values in S0 , FbtY denotes the completion of
r
σ( 0 h(Y (s))ds : r ≤ t, h ∈ B(S0 )) ∨ σ(Y (0)).
2
Martingale properties of conditional distributions
Let S be a complete, separable metric space. C(S) will denote the space of R-valued, continuous functions on S, M (S) the Borel measurable functions, Cb (S) the bounded continuous
functions, B(S) the bounded measurable functions, and P(S) the space of probability measures on S. MS [0, ∞) will denote the space of measurable functions x : [0, ∞) → S topologized by convergence in Lebesgue measure, DS [0, ∞) ⊂ MS [0, ∞) the space of S-valued,
cadlag functions with the Skorohod topology, and CS [0, ∞) ⊂ DS [0, ∞) the subspace of continuous functions. We consider martingale problems for operators satisfying the following
condition:
Condition 2.1
i) A : D(A) ⊂ Cb (S) → M (S) with 1 ∈ D(A) and A1 = 0.
ii) Either R(A) ⊂ C(S) or there exists a complete separable metric space U, a transition
2
function η from S to U, and an operator A1 : D(A) ⊂ Cb (S) → C(S × U) such that
Z
Af (x) =
A1 f (x, z)η(x, dz), f ∈ D(A).
(2.1)
U
iii) There exist ψ ∈ C(S), ψ ≥ 1, and constants af such that f ∈ D(A) implies
|Af (x)| ≤ af ψ(x),
or if A is of the form (2.1), there exist ψ1 ∈ C(S × U), ψ1 ≥ 1, and constants af such
that, for all (x, z) ∈ S × U
|A1 f (x, z)| ≤ af ψ1 (x, z).
R
(If A is of the form (2.1), then define ψ(x) ≡ U ψ1 (x, z)η(x, dz).)
iv) Defining A0 = {(f, ψ −1 Af ) : f ∈ D(A)} (or {(f, ψ1−1 A1 f ), f ∈ D(A)}), A0 is separable in the sense that there exists a countable collection {gk } ⊂ D(A) such that
A0 is contained in the bounded, pointwise closure of the linear span of {(gk , A0 gk ) =
(gk , ψ −1 Agk )} in B(S) × B(S) (or in B(S) × B(S × U)).
v) A0 is a pre-generator (for each fixed z, if A is of the form (2.1)), that is, A0 is dissipative and there are sequences of functions µn : S → P(S) and λn : S → [0, ∞) such
that for each (f, g) ∈ A
Z
(2.2)
g(x) = lim λn (x) (f (y) − f (x))µn (x, dy)
n→∞
S
for each x ∈ S.
vi) D(A) is closed under multiplication and separates points.
Remark 2.2 Suppose that we are interested in a diffusion X in a closed set S ⊂ Rd with
absorbing boundary conditions, that is,
Z t∧τ
Z t∧τ
X(t) = X(0) +
σ(X(s))dW (s) +
b(X(s))ds,
(2.3)
0
0
where τ = inf{t : X(t) ∈ ∂S} = inf{t : X(t) ∈
/ So }, where ∂S is the topological
boundary of
P1
o
>
S
and
S
is
the
interior
of
S.
Setting
a(x)
=
σ(x)σ(x)
and
Lf
(x)
=
a
(x)∂
i ∂j f (x) +
2 ij
P
2
d
bi (x)∂i f (x) for f ∈ C (R ), assuming sufficient smoothness, the natural generator would
be Lf with domain being the C 2 -functions satisfying Lf (x) = 0, x ∈ ∂S. This domain does
not satisfy Condition 2.1(vi). However, if we take D(A) = Cb2 (S), U = {0, 1}, A1 f (x, u) =
uLf (x) and η(x, du) = 1So (x)δ1 (du) + 1∂S (x)δ0 (du), where δ0 and δ1 are the Dirac measures
at 0 and 1 respectively, we have Af (x) = 1So (x)Lf (x) with domain satisfying Condition
2.1(vi). Any solution of (2.3) will be a solution of the martingale problem for A, and any
solution of the martingale problem for A will be a solution of the martingale problem for the
natural generator.
3
Definition 2.3 Let A satisfy Condition 2.1. A measurable, S-valued process X is a solution
of the martingale problem for A, if there exists a filtration {Ft } such that X is {Ft }-adapted,
Z t
(2.4)
E[ ψ(X(s))ds] < ∞, t ≥ 0,
0
and for each f ∈ D(A),
t
Z
f (X(t)) − f (X(0)) −
Af (X(s))ds
(2.5)
0
is an {Ft }-martingale. For ν0 ∈ P(S), X is a solution of the martingale problem for (A, ν0 ),
if X is a solution of the martingale problems for A and X(0) has distribution ν0 .
A measurable, S-valued process X and a nonnegative random variable τ are a solution
of the stopped martingale problem for A, if there exists a filtration {Ft } such that X is
{Ft }-adapted, τ is a {Ft }-stopping time,
Z t∧τ
ψ(X(s))ds] < ∞, t ≥ 0,
(2.6)
E[
0
and for each f ∈ D(A),
t∧τ
Z
f (X(t ∧ τ )) − f (X(0)) −
Af (X(s))ds
(2.7)
0
is an {Ft }-martingale.
A measurable, S-valued process X is a solution of the local-martingale problem for A,
if there exists a filtration {Ft } such that X is {Ft }-adapted and a sequence {τn } of {Ft }stopping times such that τn → ∞ a.s. and for each n, (X, τn ) is a solution of the stopped
martingale problem for A using the filtration {Ft }.
Remark 2.4 Note that (2.4) ensures the integrability of (2.5) and similarly for the forward
equation (2.8). Furthermore, if M f (t) denotes the process in (2.5), then (2.4) together with
Condition 2.1(iii) imply that M f is a martingale if and only if it is a local martingale, since
Z t
f
sup |M (s)| ≤ 2kf k +
ψ(X(s))ds
s∈[0,t]
0
has finite expectation for all t ≥ 0.
If X is a solution of the local martingale problem for A, then the localizing sequence {τn }
can be taken to be predictable. In particular, we can take
Z t
τn = inf{t :
ψ(X(s))ds ≥ n}.
0
Definition 2.5 Uniqueness holds for the (local) martingale problem for (A, ν0 ) if and only
if all solutions have the same finite-dimensional distributions. Stopped uniqueness holds if
for any two solutions, (X1 , τ1 ), (X2 , τ2 ), of the stopped martingale problem for (A, ν0 ), there
e and nonnegative random variables τe1 , τe2 such that (X,
e τe1 ∨e
exists a stochastic process X
τ2 ) is a
e
solution of the stopped martingale problem for (A, ν0 ), (X(·∧e
τ1 ), τe1 ) has the same distribution
e
as (X1 (· ∧ τ1 ), τ1 ), and (X(· ∧ τe2 ), τe2 ) has the same distribution as (X2 (· ∧ τ2 ), τ2 ).
4
Remark 2.6 Note that stopped uniqueness implies uniqueness. Stopped uniqueness holds
if uniqueness holds and every solution of the stopped martingale problem can be extended
(beyond the stopping time) to a solution of the (local) martingale problem. (See Lemma
4.5.16 of Ethier and Kurtz (1986) for conditions under which this extension can be done.)
Definition 2.7 A P(S)-valued
function {νt , t ≥ 0} is a solution of the forward equation
Rt
for A if for each t > 0, 0 νs ψds < ∞ and for each f ∈ D(A),
Z
νt f = ν0 f +
t
νs Af ds.
(2.8)
0
A pair of measure-valued functions {(νt0 , νt1 ), t ≥ 0} is a solution
R t 1 of the stopped forward
0
1
equation for A if for each t ≥ 0, νt ≡ νt + νt ∈ P(S) and 0 νs ψds < ∞, t → νt0 (C) is
nondecreasing for all C ∈ B(S), and for each f ∈ D(A),
Z t
νs1 Af ds.
νt f = ν0 f +
(2.9)
0
A P(S)-valued function {νt , t ≥ 0} is a solution of the local forward equation for A if
there exists a sequence {(ν 0,n , ν 1,n )} of solutions of the stopped forward equation for A such
that for each C ∈ B(S) and t ≥ 0, {νt1,n (C)} is nondecreasing and limn→∞ νt1,n (C) = νt (C).
Clearly, any solution X of the martingale problem for A gives a solution of the forward equation for A, that is νt f = E[f (X(t)], and any solution of the stopped martingale problem for A gives a solution of the stopped forward equation for A, that is,
νt0 f = E[1[τ,∞) (t)f (X(τ ))] and νt1 f = E[1[0,τ ) (t)f (X(t))]. The primary consequence of Condition 2.1 is the converse.
Lemma 2.8 If A satisfies Condition 2.1 and {νt , t ≥ 0} is a solution of the forward equation
for A, then there exists a solution X of the martingale problem for A satisfying νt f =
E[f (X(t))].
If A satisfies Condition 2.1 and {(νt0 , νt1 ), t ≥ 0} is a solution of the stopped forward
equation for A, then there exists a solution (X, τ ) of the stopped martingale problem for A
such that νt0 f = E[1[τ,∞) (t)f (X(τ ))] and νt1 f = E[1[0,τ ) (t)f (X(t))].
If A satisfies Condition 2.1 and {νt , t ≥ 0} is a solution of the local forward equation
for A, then there exists a solution X of the local martingale problem for A satisfying νt f =
E[f (X(t)].
Proof. Various forms of the first part of this result exist in the literature beginning with
the result of Echeverrı́a (1982) for the stationary case, that is, νt ≡ ν0 and ν0 Af = 0.
Extension of Echeverria’s result to the forward equation is given in Theorem 4.9.19 of Ethier
and Kurtz (1986) for locally compact spaces and in Theorem 3.1 of Bhatt and Karandikar
(1993) for general complete separable metric spaces. The version given here is a special case
of Corollary 1.12 of Kurtz and Stockbridge (2001).
The the result for stopped forward equations also follows by the same corollary. First
e =
enlarge the state space e
S = S × {0, 1} and define νet h = νt0 h(·, 0) + νt1 h(·, 1). Setting D(A)
5
e define Ah(x,
e
{f (x)g(y) : f ∈ D(A), g ∈ B({0, 1})}, for h = f g ∈ D(A),
y) = yAh(x, y) =
yg(y)Af (x) and Bh(x, y) = y(h(x, 0) − h(x, y)). Then
Z t
0
1
0
1
e
0 = νt h(·, 1) + νt h(·, 1) − ν0 h(·, 1) − ν0 h(·, 1) −
νes Ahds
0
Z t
e
= νet h − νe0 h −
νes Ahds
+ νt0 h(·, 1) − νt0 h(·, 0) + ν00 h(·, 1) − ν00 h(·, 0)
Z0 t
Z
e
= νet h − νe0 h −
νes Ahds −
Bh(x, y)µ(dx × dy × ds),
0
S×{0,1}×[0,t]
where, noting that νt0 (C) is an increasing function of t, µ is the measure determined by
µ(C × {1} × [0, t2 ]) = νt02 (C) − ν00 (C),
µ(C × {0} × [t1 , t2 ]) = 0.
e Y)
Corollary 1.12 of Kurtz and Stockbridge (2001) then implies the existence of a process (X,
0
1
e
e
in (S × {0, 1}) such that νt f = E[(1 − Y (t))f (X(t))],
νt f = E[Y (t)f (X(t))],
and
Z t
e
e
e
f (X(t)) − f (X(0)) −
Y (s)Af (X(s))ds
0
is a martingale for each f ∈ D(A). Following the arguments in Section 2 of Kurtz and
Stockbridge (2001), the process can be constructed in such a way that Y (s) = 0 implies
e = X(τ
e ) for t ≥ τ .
Y (t) = 0 for t > s, and hence τ = inf{t : Y (t) = 0}. Note that X(t)
Similarly, suppose {(ν 0,n , ν 1,n )} is the sequence of solutions of the stopped forward equation associated with a solution of the local forward equation and take (νt0,0 , νt1,0 ) ≡ (ν0 , 0).
For f ∈ B(S × Z+ ), define
νbt f =
∞
X
(νt1,n f (·, n) − νt1,n−1 f (·, n))
n=1
and
Z
f (x, n)b
µ(dx × dn × ds) =
S×Z+ ×[0,t]
∞
X
νt0,n f (·, n).
n=1
Note that νbt is a probability measure with S-marginal νt = limn→∞ νt1,n .
Setting
e = D(B) = {gf : g ∈ Cc (Z+ ), f ∈ D(A)}
D(A)
(where, of course, Cc (Z+ ) is the collection of functions with finite support) and defining
e (x, n) = g(n)Af (x) and Bgf (x, n) = f (x)(g(n + 1) − g(n)),
Agf
Z t
Z
e
νbt gf = νb0 gf +
νbs Agf ds +
Bgf db
µ.
S×Z+ ×[0,t]
0
Let 0 < ψ0 (n) < 1 satisfy
X
n
Z
ψ0 (n)
n
νs1,n ψds < ∞.
0
6
e n) = ψ(x)ψ0 (n), and
e satisfies Condition (2.1) with ψ replaced by ψ(x,
Then A
Z t
e < ∞, t > 0.
νbs ψds
0
Corollary 1.12 of Kurtz and Stockbridge (2001) then implies the existence of a process (X, N )
such that (X(t), N (t)) has distribution νbt and a random measure Γ on S × Z+ × [0, ∞)
satisfying
Z
Z
E[
f (x, n)Γ(dx × dn × ds)] =
f (x, n)b
µ(dx × dn × ds)
S×Z+ ×[0,t]
S×Z+ ×[0,t]
e
such that for each gf ∈ D(A),
Z t
Z
g(N (t))f (X(t)) −
g(N (s))Af (X(s))ds −
f (x)Bg(n)Γ(dx × dn × ds) (2.10)
S×Z+ ×[0,t]
0
is a {FtX,N }-martingale.
Rt
Let τk = inf{t : 0 ψ(X(s))ds ≥ k}. Let gm (n) = 1[0,m] (n), and consider the limit of the
sequence of martingales
Z t∧τk
gm (N (s))Af (X(s))ds
(2.11)
gm (N (t ∧ τk ))f (X(t ∧ τk )) −
0
Z
−
f (x)Bgm (n)Γ(dx × dn × ds)
S×Z+ ×[0,t∧τk ]
as m → ∞. The first two terms converge in L1 by the dominated convergence theorem, and
the third term satisfies
Z
f (x)Bgm (n)Γ(dx × dn × ds)|] ≤ kf kb
µ(S × {m} × [0, t])
E[|
S×Z+ ×[0,t∧τk ]
= kf kνt0,m (S)
and hence converges to zero in L1 . It follows that
Z t∧τk
f (X(t ∧ τk )) −
Af (X(s))ds
0
is a martingale, and consequently, X is a solution of the local martingale problem for A such
that X(t) has distribution νt .
Let X be a solution of the martingale problem for A with respect to a filtration {Ft },
and let {Gt } be a filtration with Gt ⊂ Ft . Then letting πt denote the conditional distribution
of X(t) given Gt , Lemma A.1 implies that for each f ∈ D(A),
Z t
πt f − π 0 f −
πs Af ds
(2.12)
0
7
is a {Gt }-martingale.
Let S0 and S0 be complete, separable metric spaces, and let γ : S → S0 be Borel measurable. Let X be a solution of the martingale problem for A, and let Z be a S0 -valued random
variable. Assume that Ft ⊃ σ(Z) for all t ≥ 0. Define Y (t) = γ(X(t)),
Z r
Y
g(Y (s))ds : r ≤ t, g ∈ B(S0 )) ∨ σ(Y (0),
(2.13)
Fbt = completion of σ(
0
FbtY,Z = FbtY ∨ σ(Z), πt (C) = P{X(t) ∈ C|FbtY,Z }, where by Theorem A.3 of Kurtz (1998),
we can assume that π is a progressively measurable, P(S)-valued process and πt∧τ (C) =
Y,Z
P{X(t ∧ τ ) ∈ C|Fbt∧τ
} for every {FbtY,Z }-stopping time τ .
Remark 2.9 If Y is cadlag with no fixed points of discontinuity, then by Lemma A.5, FtY =
FbtY .
Note that
Z
t
Z
t
πs (g ◦ γ)ds =
0
for each g ∈ B(S0 ),
g(Y (s))ds,
(2.14)
0
and if (2.4) holds,
t
Z
πt f −
πs Af ds
0
is a {FbtY,Z }-martingale for each f ∈ D(A). With these properties in mind, we work with a
definition of the filtered martingale problem slightly more general than that of Kurtz (1998).
e τe) ∈ MS0 [0, ∞) × MP(S) [0, ∞) × S0 × [0, ∞]
Definition 2.10 Let µ
b0 ∈ P(S × S0 ). (Ye , π
e, Z,
is a solution of the stopped, filtered martingale problem for (A, γ, µ
b0 ), if
e =µ
E[e
π0 (C)1D (Z)]
b0 (C × D),
π
e is {FbtY Z }-adapted, τe is a {FbtY ,Z }-stopping time, for each g ∈ B(S0 ) and t ≥ 0,
Z t
Z t
π
es (g ◦ γ)ds =
g(Ye (s))ds ,
ee
(2.15)
e e
0
(2.16)
0
Z
E[
0
t∧e
τ
π
es ψds] < ∞, t > 0,
(2.17)
and for each f ∈ D(A),
Z
ff (t ∧ τe) ≡ π
M
et∧eτ f −
0
t∧e
τ
π
es Af ds
(2.18)
is a {FbtY Z }-martingale.
e τe) satisfies all the conditions except (2.15), we will refer to it as a solution of
If (Ye , π
e, Z,
the stopped, filtered martingale problem for (A, γ).
e is a solution of the filtered martingale problem for (A, γ).
If τe = ∞ a.s., then (Ye , π
e, Z)
e ∈ MS0 [0, ∞) × MP(S) [0, ∞) × S0 is a solution of the filtered local-martingale
(Ye , π
e, Z)
e e
problem for (A, γ) if there exists a sequence {e
τn } of {FbtY ,Z }-stopping times such that τen → ∞
e τen ) is a solution of the stopped, filtered martingale problem for
a.s. and for each n, (Ye , π
e, Z,
(A, γ).
ee
8
Remark 2.11 By the optional projection theorem (see Theorem A.3 of Kurtz (1998)), there
e e
exists a modification of π
e such that for all t ≥ 0 and all {FbtY ,Z }-stopping times τ , π
et∧τ is
e ,Z
e
Y
Fbt∧τ -measurable. Consequently, we will assume that π
e has this property.
Remark 2.12 Kurtz and Ocone (1988) consider the filtered martingale problem with S =
S1 × S0 and γ the projection onto S0 .
The following lemma is an immediate consequence of (2.16).
e τe) is a solution of the stopped, filtered martingale problem for
Lemma 2.13 If (Ye , π
e, Z,
e
(A, γ, µ
b0 ), then FbtY is contained in the completion of σ(e
πs , s ≤ t) ∨ σ(Ye (0)).
If X is a solution of the martingale problem for A, then X (r) given by X (r) (t) = X(r + t)
is also a solution of the martingale problem for A. The following lemma gives the analogous
result for filtered martingale problems. Let Yer denote the restriction of Ye to [0, r].
e τe) ∈ MS0 [0, ∞) × MP(S) [0, ∞) × S0 × [0, ∞] is a solution
Lemma 2.14 Suppose (Ye , π
e, Z,
e e
of the stopped, filtered martingale problem for (A, γ). For r ≥ 0 such that Ye (r) is FbrY ,Z e Yer ) ∈ S0 × M [0, r], and π
measurable, let Yb (t) = Ye (r + t), Zb = (Z,
bt = π
er+t . Then
b (e
(Yb , π
b, Z,
τ − r) ∨ 0) ∈ MS0 [0, ∞) × MP(S) [0, ∞) × S0 × M [0, r] × [0, ∞]
is a solution of the stopped, filtered martingale problem for (A, γ).
e is a solution of the filtered martingale problem
Suppose τe = ∞, a.s. (that is, (Ye , π
e, Z)
e e
er = π
b0 ,
for (A, γ)). For r ≥ 0 such that Ye (r) is FbrY ,Z -measurable, let Yb (t) = Y (r + t), Zb = π
and
b
π
bt = E[e
πr+t |FbtY ∨ σ(Y (r)) ∨ σ(e
πr )].
b ∈ MS0 [0, ∞) × MP(S) [0, ∞) × P(S) is a solution of the filtered martingale
Then (Yb , π
b, Z)
problem for (A, γ).
Proof. In the second part of the lemma, the existence of π
b as an adapted, P(S)-valued
process follows by Theorem A.3 of Kurtz (1998) and
Z t
Z r+t
E[ π
bs ψds] = E[
π
es ψds] < ∞.
0
r
In both parts, the required martingale properties follow by Lemma A.1.
3
Conditional distributions and solutions of martingale
problems
Of course, the forward equation is a special case of (2.12) in which the “martingale” is
identically zero. Consequently, the following proposition can be viewed as an extension of
Lemma 2.8.
9
Proposition 3.1 Let A satisfy Condition 2.1. Suppose that Ye is a cadlag, S0 -valued process
e
no fixed points of discontinuity, {e
πt , t ≥ 0} is a P(S)-valued process, adapted to {FtY },
Rwith
t
π
e ψds < ∞ a.s., t ≥ 0, and
0 s
Z
π
et f − π
e0 f −
t
π
es Af ds
0
(3.1)
is a {FtY }-local-martingale for each f ∈ D(A). Then there exist a solution X of the local
martingale problem for A, a cadlag, S0 -valued process Y , and a P(S)-valued process {πt , t ≥
0} such that (Y, π) has the same finite-dimensional distributions as (Ye , π
e) and πt is the
Y
conditional distribution of X(t) given Ft .
For each t ≥ 0, there exists a Borel measurable mapping Ht : DS0 [0, ∞) → P(S) such
that πt = Ht (Y ) and π
et = Ht (Ye ) almost surely.
e
Proof. As in Kurtz (1998), we begin by enlarging the state space so that the current
state of the process contains all information about the past of the observation Ye . Let
{bk }, {ck } ⊂ Cb (S0 ) satisfy 0 ≤ bk , ck ≤ 1, and suppose that the spans of {bk } and {ck } are
bounded, pointwise dense in B(S0 ). (Existence of {bk } and {ck } follows from the separability
of S0 .) Let a1 , a2 , . . . be an ordering of the rationals with ai ≥ 1. For k, i ≥ 1, let
Z t
Z t
eki (t) = ck (Ye (0)) − ai
eki (s)ds +
U
U
bk (Ye (s))ds
(3.2)
0
0
Z t
−ai t
e
= ck (Y (0))e
+
e−ai (t−s) bk (Ye (s))ds.
0
e1i (t) = e−ai t and U
e1i (t) determines the value of t. Let
If we assume c1 = 1 and b1 = 0, U
∞
e (t) = (U
eki (t) : k, i ≥ 1) ∈ [0, 1] .
U
Define F : (r, y) ∈ [0, ∞) × DS0 [0, ∞) → u ∈ [0, 1]∞ by
Z r
−ai r
uki (r, y) = ck (y(0))e
+
e−ai (r−s) bk (y(s))ds,
0
e (t) = F (t, Ye ). Properties of Laplace transforms and the assumption that Ye has no fixed
so U
points of discontinuity imply that there are measurable mappings Λ : [0, 1]∞ → DS0 [0, ∞)
e (t)) = Ye (· ∧ t) and Λ0 (U
e (t)) = Ye (t) almost surely. We
and Λ0 : [0, 1]∞ → S0 such that Λ(U
−ai t
can define Λ so that if u1,i = e , then y = Λ(u) satisfies y(s) = y(t−) for s ≥ t. Note that
these observations imply that
e (t)) = FbtYe = FtYe ,
the completion of σ(U
where the second equality follows by Lemma A.5.
b be the collection of functions on b
Let b
S = S × [0, 1]∞ , and let D(A)
S given by
{f (x)
m
Y
gki (uki ) : f ∈ D(A), gki ∈ C 1 [0, 1], m = 1, 2, . . .}.
k,i=1
10
Writing g(u) instead of
b
by ∂ki g, for f g ∈ D(A),
Q
ki
gki (uki ) and denoting the partial derivative with respect to uki
e (t)) − π
e (0))
π
et f g(U
e0 f g(U
Z t
X
e (s))e
eki (s) + bk (Ye (s)))∂ki g(U
e (s)) ds
g(U
πs Af + π
es f
(−ai U
−
0
e
is a {FtY }-local-martingale.R Note that
t
sequence to be τen = inf{t : 0 π
es ψds ≥
Define
without loss of generality, we can take the localizing
n}.
X
b1 (f g)(x, u, z) = g(u)Af (x) + f (x)
A
(−ai u + bk (z))∂ki g(u),
and
Z
b g)(x, u) =
A(f
b1 (f g)(x, u, z)η(x, u, dz),
A
(3.3)
where, with reference to (2.1) and the definition of Λ0 , we define η(x, u, dz) = δΛ0 (u) (dz).
Define νet0,n , νet1,n ∈ M(S × [0, 1]∞ ) by
Z
0,n
e (e
νet h = E[1[eτn ,∞) (t) h(z, U
τn ))e
πτen (dz)]
S
Z
1,n
e (t))e
πt (dz)] .
νet h = E[1[0,eτn ) (t) h(z, U
S
b
Setting νetn = νet0,n + νet1,n , for f g ∈ D(A),
e (t ∧ τen ))]
νetn (f g) = E[e
πt∧eτn f g(U
e (0))]
= E[e
π0 f g(U
R t∧eτn P
e
e
e
e
+E[ 0
g(U (s))e
πs Af + π
es f (−ai Uki (s) + bk (Y (s)))∂ki g(U (s)) ds]
R t 1,n
b g)ds.
= νe0 (f g) + νe A(f
0
s
b and νe =
Consequently, (e
ν 0,n , νe1,n ) is a solution of the stopped forward equation for A,
1,n
limn→∞ νe is a solution of the local forward equation. By Lemma 2.8, there exists a solution
b such that
(X, U ) of the local martingale problem for A,
E[f (X(t))
m
Y
gki (Uki (t))] = νet (f
k,i=1
m
Y
gki )
k,i=1
m
Y
= E[e
πt f
(3.4)
eki (t))] .
gki (U
k,i=1
e (t) have the same distribution. If we define Y (· ∧ t) =
It follows that for each t, U (t) and U
e
Λ(U (t)), Y (· ∧ t) and Y (· ∧ t) have the same distribution on DS0 [0, ∞).
Define πt as the conditional distribution of X(t) given FtY . Then, for any bounded
measurable function g on [0, 1]∞
E[f (X(t))g(U (t)] = E[πt f g(U (t))]
e (t))] .
= E[e
πt f g(U
11
(3.5)
e
e (t)), for every t,
Since FtY is the completion of σ(U (t)) and FtY is the completion of σ(U
et : [0, 1]∞ → P(S) such that πt = Gt (U (t)) a.s. and π
et (U
e (t))
there exist mappings Gt , G
et = G
a.s. By (3.4),
et (U
e (t))f h(U
e (t))]
E[Gt (U (t))f h(U (t))] = E[G
et (U (t))f h(U (t))]
= E[G
(3.6)
e (t)
for all h ∈ B(S0 × [0, 1]∞ ), where the last equality follows from the fact that U (t) and U
et (·)f , we have
have the same distribution. Applying (3.6) with h = Gt (·)f and with h = G
2
e
e
E[Gt (U (t))f Gt (U (t))f ] = E[ Gt (U (t))f ) ] = E[(Gt (U (t))f ))2 ],
and it follows that
2
et (U (t))f ] = 0 .
E[ Gt (U (t))f − G
e (t))f a.s., and hence (πt , U (t)) has the same distribution as
Consequently, π
et f = Gt (U
e (t)).
(e
πt , U
e (t)) determines U (s) (U
e (s)) for s < t, U and U
e have the same distribution on
Since U (t) (U
C[0,1]∞ [0, ∞). Consequently, (π, Y ) and (e
π , Ye ) have the same finite-dimensional distributions.
The mapping Ht is given by Ht (y) ≡ Gt (F (t, y)).
Corollary 3.2 Let A satisfy Condition 2.1. Suppose that {e
πt , t ≥ 0} is a cadlag, P(S)e
valued
R tprocess with no fixed points of discontinuity adapted to a complete filtration {Gt } such
that 0 π
es ψds < ∞ a.s., t ≥ 0, and
Z
π
et f − π
e0 f −
0
t
π
es Af ds
is a {Get }-local martingale for each f ∈ D(A). Then there exists a solution X of the local
martingale problem for A, a P(S)-valued process {πt , t ≥ 0} such that {πt , t ≥ 0} has
the same distribution as {e
πt , t ≥ 0}, and a filtration {Gt } such that πt is the conditional
distribution of X(t) given Gt .
Proof. The corollary follows by taking Ye (t) = π
et and applying Proposition 3.1.
The next corollary extends Corollary 3.5 of Kurtz (1998).
Corollary 3.3 Let A satisfy Condition 2.1. Let γ : S → S0 be Borel measurable, and let
α be a transition function from S0 into S (y ∈ S0 → α(y, ·) ∈ P(S) is Borel measurable)
R
e
satisfying α(y, γ −1 (y)) = 1. Assume that ψ(y)
≡ S ψ(z)α(y, dz) < ∞ for each y ∈ S0 and
define
Z
Z
C = {( f (z)α(·, dz), Af (z)α(·, dz)) : f ∈ D(A)} .
S
Let µ0 ∈ P(S0 ), and define ν0 =
S
R
α(y, ·)µ0 (dy).
12
Rt
e Ye (s))ds <
a) If Ye is a solution of the local-martingale problem for (C, µ0 ) satisfying 0 ψ(
∞ a.s., then there exists a solution X of the local-martingale problem for (A, ν0 ) such
that Ye has the same distribution on MS0 [0, ∞) as Y = γ ◦ X. If Y and Ye are cadlag,
then Y and Ye have the same distribution on DS0 [0, ∞).
b) If Y (t) is FbtY -measurable (which by Lemma A.4 holds for almost every t), then α(Y (t), ·)
is the conditional distribution of X(t) given FbtY .
c) If, in addition, uniqueness holds for the martingale problem for (A, ν0 ), then uniqueness
holds for the MS0 [0, ∞)-martingale problem for (C, µ0 ). If Ye has sample paths in
DS0 [0, ∞), then uniqueness holds for the DS0 [0, ∞)-martingale problem for (C, µ0 ).
d) If uniqueness holds for the martingale problem for (A, ν0 ), then Y restricted to TY =
{t : Y (t) is FbtY -measurable} is a Markov process.
Proof. We are not assuming that Ye is cadlag, so to apply Proposition 3.1, replace Ye by the
e given by (3.2). Observing that FtUe = FbtYe , define
continuous process U
e
e
π
et = E[α(Ye (t), ·)|FtU ] = E[α(Ye (t), ·)|FbtY ],
and note that π
et = α(Y (t), ·) for t ∈ TY . Then
Z t
Z t
αAf (Ye (s))ds
π
es Af ds = π
et f − αf (Ye (0)) −
π
et f − π
e0 f −
e
0
0
is a {FtU }-local martingale for each f ∈ D(A) and Proposition 3.1 gives the existence of the
processes X and U such that X is a solution of the local-martingale problem for A and πt ,
et . Consequently, for
the conditional distribution of X(t) given FtU has the distribution as π
almost every t, πt = α(γ(X(t)), ·) and it follows that Y = γ ◦ X has the same distribution
on MS0 [0, ∞) as Ye .
Since the finite-dimensional distributions of X are uniquely determined, the distribution
of γ ◦ X (and hence of Ye ) on MS0 [0, ∞) is uniquely determined. If Ye has sample paths in
DS0 [0, ∞), then its distribution on DS0 [0, ∞) is determined by its distribution on MS0 [0, ∞).
Since X is the unique solution of a martingale problem, by Lemma A.13, it is Markov.
The Markov property for Y for t ∈ TY follows from the Markov property for X by
e
E[f (Y (t + s))|FbtY ] = E[E[f (γ(X(t + s)))|FtX ]|FbtY ]
= E[hf,t,s (X(t))|FbtY ]
Z
=
hf,t,s (x)α(Y (t), dx).
S
We will also need a stopped version of Proposition 3.1.
13
Proposition 3.4 Let A satisfy Condition 2.1. Suppose that Ye is a cadlag, S0 -valued process
e
with no fixed points of discontinuity, τe is a {FtY }-stopping time, {e
πt , t ≥ 0} is a P(S)-valued
R
t∧τ
Ye
process, adapted to {Ft }, 0 π
es ψds < ∞ a.s., t ≥ 0, and
Z
π
et∧eτ f − π
e0 f −
0
t∧e
τ
π
es Af ds
(3.7)
is a {FtY }-martingale for each f ∈ D(A). Then there exist a solution (X, τ ) of the stopped
martingale problem for A, a cadlag, S0 -valued process Y , and a P(S)-valued process {πt , t ≥
0} such that {(Y (t ∧ τ ), πt 1{τ ≥t} ), t ≥ 0} has the same distribution as {(Ye (· ∧ τe), π
et 1{eτ ≥t} )
Y
and πt∧τ is the conditional distribution of X(t) given Ft∧τ .
e
b and U
e defined as in the proof or Proposition 3.1,
Proof. With A
Z
0
e (e
τ ))e
πτe(dz)]
νet h = E[1[eτ ,∞) (t) h(z, U
S
Z
1
e (t))e
νet h = E[1[0,eτ ) (t) h(z, U
πt (dz)]
S
b Lemma 2.8 ensures the existence
defines a solution of the stopped martingale problem for A.
b such that
of a solution (X, U, τ ) of the stopped martingale problem for A
e (t ∧ τe))].
E[f (X(t ∧ τ ))g(U (t ∧ τ ))] = E[e
πt∧eτ f g(U
e (t ∧ τe) and hence
Then for t ≥ 0, U (t ∧ τ ) has the same distribution as U
Y (· ∧ τ ) = lim Y (· ∧ t ∧ τ ) ≡ lim Λ(U (t ∧ τ ))
t→∞
t→∞
has the same distribution as Ye (· ∧ τe). With reference to Section A.3,
Y
E[f (X(t ∧ τ ))|σ(U (t ∧ τ ))] = E[f (X(t ∧ τ ))|Gt∧τ
]
has the same distribution as
Ye
e (t ∧ τe))] = E[e
E[e
πt∧eτ f |σ(U
πt∧eτ f |Gt∧e
τ ],
and by Lemma A.10, π
et 1{eτ ≥t} has the same distribution as πt 1{τ ≥t} , where πt is the condiY
tional distribution of X(t ∧ τ ) given Ft∧τ
.
The only place that Condition 2.1 is used in the proof of Proposition 3.1 is to conclude
b defined in (3.3) corresponds to a
that every solution of the local forward equation for A
b can be
solution of the local martingale problem. For the filtered martingale problem, A
given explicitly by
X
b g)(x, u) = g(u)Af (x) + f (x)
A(f
(−ai u + bk ◦ γ(x))∂ki g(u),
(3.8)
Consequently, we state the next result under the following hypothesis.
14
b defined by (3.8), each solution of the local forward equation for A
b
Condition 3.5 For A
b
corresponds to a solution of the local martingale problem for A.
We have the following generalization of Theorem 3.2 of Kurtz (1998).
Theorem 3.6 Let A ⊂ B(S) × M (S), µ
b0 ∈ P(S × S0 ), and γ : S → S0 be Borel measurable,
e
e
and assume Condition 3.5. Let (Y , π
e, Z) be a solution of the local filtered martingale problem
for (A, γ, µ
b0 ). Then the following hold:
a) There exists a solution X of the local-martingale problem for A and an S0 -valued random variable Z such that (X(0), Z) has distribution µ
b0 and Y = γ ◦ X has the same
e
distribution on MS0 [0, ∞) as Y .
b) Let πt be the conditional distribution of X(t) given FbtY,Z . For each t ≥ 0, there exists
a Borel measurable mapping Ht : MS0 [0, ∞) × S0 → P(S) such that πt = Ht (Y, Z) and
e
π
et = Ht (Ye , Z).
c) If Y and Ye have sample paths in DS0 [0, ∞), then Y and Ye have the same distribution
on DS0 [0, ∞) and Ht is a Borel measurable mapping from DS0 [0, ∞) × S0 to P(S).
d) If uniqueness holds for the local martingale problem for (A, ν0 ), then uniqueness holds
for the filtered local-martingale problem for (A, γ, µ
b0 ) in the sense that if (Y, π, Z) and
e
e
(Y , π
e, Z) are solutions, then for each 0 ≤ t1 < · · · < tm , (πt1 , . . . , πtm , Y, Z) and
e have the same distribution on P(S)m × MS0 [0, ∞) × S0 .
(e
πt1 , . . . , π
etm , Ye , Z)
Remark 3.7 Note that the theorem does not assume that γ is continuous.
e in (3.2), replace ck (Ye (0)) by ck (Ye (0), Z).
e Note that for a.e. t,
Proof. In the definition of U
f1 (Ye (t ∧ τe))e
πt∧eτ f2 = π
et∧eτ (f2 f1 ◦ γ)
a.s.
(3.9)
(First consider f1 = 1C , C ∈ B(S0 ).)
With νetn,0 and νetn,1 defined as before,
e (t ∧ τen ))]
νetn (f g) = E[e
πt∧eτn f g(U
e (0))]
= E[e
π0 f g(U
R t∧eτn P
e (s))e
eki (s) + bk (Ye (s)))∂ki g(U
e (s)) ds]
+E[ 0
g(U
πs Af + π
es f (−ai U
e (0))]
= E[e
π0 f g(U
R t∧eτn P
e (s))e
eki (s) + π
e (s)) ds]
+E[ 0
g(U
πs Af + (−ai U
es (f bk ◦ γ))∂ki g(U
Rt
b g)ds,
= νe0 (f g) + νe1,n A(f
0
s
b is defined in (3.8).
where the third equality follows from (3.9) and A
e
We are not assuming that Y is cadlag, but we still conclude that the completion of
e (t)) is FbtYe ,Ze and there exist Λ : [0, 1]∞ → MS0 [0, ∞) × S0 and Λ1 : [0, 1]∞ → S0 such that
σ(U
15
e (t)) = (Ye (· ∧ t), Z)
e and Λ1 (U
e (0)) = Z.
e Condition 3.5 ensures the existence of a solution
Λ(U
b such that U and U
e have the same distribution,
(X, U ) of the local martingale problem for A
Z t
Z t
bk (Y (s))ds
(3.10)
Uki (s)ds +
Uki (t) = Uki (0) − ai
0
0
Z t
−ai t
e−ai (t−s) bk (Y (s))ds,
= Uki (0)e
+
0
and defining Z = Λ1 (U (0)), Parts (a) and (b) hold.
Part (c) follows from the fact that the distribution of a cadlag process is determined by
its distribution on MS0 [0, ∞).
Finally, for Part (d), uniqueness for the local martingale problem for (A, ν0 ) implies
b νb0 ), where νb0 h = E[e
e (0))]. Uniqueuniqueness for the local martingale problem for (A,
π0 h(·, U
ness of the distribution of (X, U ) in Part (b) implies uniqueness of the distribution of
e
(Ye , π
e, Z).
3.1
The Markov property
Uniqueness for martingale problems usually implies the Markov property for solutions, and
a similar result holds for filtered martingale problems.
Theorem 3.8 Let A ⊂ B(S) × M (S), µ
b0 ∈ P(S × S0 ), and γ : S → S0 be Borel measurable,
e
e
and assume Condition 3.5. Let (Y , π
e, Z) be a solution of the filtered martingale problem for
(A, γ, µ
b0 ). (e
τ = ∞.) If uniqueness holds for the martingale problem for (A, ν0 ), then π
e is a
P(S)-valued Markov process.
Proof. Fix r ≥ 0, and let (Yb , π
b) be as in the second part of Lemma 2.14. Since π
b0 = π
er ,
∗
∗
∗
they have the same distribution. By Lemma A.12, a process (Y , π , Z ) can be constructed
∗
b, π
b0 ),
, πr∗ ) has the same distribution on MS0 ×P(S) [0, ∞) × P(S) as (Yb , π
so that (Y ∗ (r + ·), πr+·
∗
∗
∗
∗
e
(Y (· ∧ r), π·∧r , Z , πr ) has the same distribution on MS0 ×P(S) [0, r] × S0 × P(S) as (Y (· ∧
e π
r), π
e·∧r , Z,
er ), and
∗
∗
∗
E[g(Y ∗ (r + ·), πr+·
)|Y ∗ (· ∧ r), π·∧r
, Z ∗ , πr∗ ] = E[g(Y ∗ (r + ·), πr+·
)|πr∗ ].
(3.11)
We claim that (Y ∗ , π ∗ , Z ∗ ) is a solution of the filtered martingale problem for (A, γ, µ
b0 ).
∗
∗
∗
∗
e
(π0 , Z ) has the same distribution as (e
π0 , Z), so (2.15) holds. Since (Y (· ∧ r), π·∧r ) has
e
the same distribution as (Y (· ∧ r), π
e·∧r ), for g ∈ B(S0 ) and t ≤ r,
Z t
Z t
∗
πs (g ◦ γ)ds =
g(Y ∗ (s))ds a.s.
0
0
Rt
Rt
R t−r
R t−r
For t > r, ( r πs∗ (g◦γ)ds, r g(Y ∗ (s))ds) has the same distribution as ( 0 π
bs (g◦γ)ds, 0 g(Yb (s))ds),
Rt
Rt
so r πs∗ (g ◦ γ)ds = r g(Y ∗ (s))ds a.s. Consequently, (2.16) follows.
16
Rt
For f ∈ D(A), let Mf∗ (t) = πt∗ f − 0 πs∗ Af ds. For r ≤ t < t + h, let H1 be a bounded
random variable measurable with respect to the completion of
Z u
σ(
h(Y ∗ (s))ds : r ≤ u ≤ t, h ∈ B(S0 )) ∨ σ(πr∗ )
r
∗
∗
∗
and H2 a bounded random variable measurable with respect to FbrY ,Z = FbrY ∨ σ(Z ∗ ). Then
by (3.11),
E[(Mf∗ (t + h) − Mf∗ (t))H1 H2 ] = E[(Mf∗ (t + h) − Mf∗ (t))H1 E[H2 |πr∗ ]],
∗
and the right side is zero by the fact that (Y ∗ (r +·), πr+·
) has the same distribution as (Yb , π
b).
It follows that
∗
∗
E[Mf∗ (t + h) − Mf∗ (t)|FbtY ,Z ] = 0.
(3.12)
∗
If t < t + h ≤ r, then (3.12) follows from the fact that (Y ∗ (· ∧ r), π·∧r
, Z ∗ , πr∗ ) has the same
e π
distribution as (Ye (· ∧ r), π
e·∧r , Z,
er ), and for t < r < t + h,
E[Mf∗ (t + h) − Mf∗ (t)|FbtY
∗ ,Z ∗
] = E[Mf∗ (t + h) − Mf∗ (r) + Mf∗ (r) − Mf∗ (t)|FbtY
∗ ,Z ∗
]=0
verifying (2.18).
e have the same distribution. Consequently,
By uniqueness, (π ∗ , Y ∗ , Z ∗ ) and (e
π , Ye , Z)
(3.11) implies
e π
E[g(Ye (r + ·), π
er+· )|Ye (· ∧ r), π
e·∧r , Z,
er ] = E[g(Ye (r + ·), π
er+· )|e
πr ],
giving the Markov property.
In the classical setting, X = (X1 , Y ) ∈ S1 × S0 and γ(X) = Y , πt = πt1 × δY (t) , where
πt1 is the conditional distribution of X1 (t) given FtY . In the observations in additive white
noise setting, a number of authors (Kunita (1971); Bhatt, Budhiraja, and Karandikar (2000);
Stettner (1989)) have given conditions under which π 1 is Markov. The following example
shows that the conclusion does not hold in general, and hence the Markov property for π 1
does not immediately follow from Theorem 3.8.
Example 3.9 Let (X1 , Y ) be the Markov process with values in {−1, +1} × N and generator
Af (x, y) = λ y[f (−x, y + 1) − f (x, y)] + µ[f (−x, y) − f (x, y)].
Given a pure jump Markov counting process (a Yule process) Y with intensity λY and an
independent Poisson process Z with intensity µ, the process (X1 , Y ) can be represented by
X1 (t) = (−1)Y (t)−Y (0)+Z(t) X1 (0).
Then the conditional distribution of X1 (t) given FtY is
πt1 (dx) = 1E (Y (t) − Y (0)) αt δ+1 (dx) + (1 − αt )δ−1 (dx)
+1O (Y (t) − Y (0)) (1 − αt )δ+1 (dx) + αt δ−1 (dx) ,
17
where E is the set of even integers, O the set of odd integers, and
αt =
1 + (2α0 − 1)e−2µt
,
2
where α0 = P{X1 (0) = 1|Y (0)}.
If α0 = 12 , then αt = 12 and πt1 (dx) = 12 δ+1 (dx) + 21 δ−1 (dx), for all t ≥ 0, and π 1 is
trivially Markov; however, if P{α0 6= 12 } > 0, in general π 1 is not Markov. Assuming,
for example, that Y (0) = 1 and α0 = P{X1 (0) = 1|Y (0)} = P{X1 (0) = 1} 6= 21 , then
1
Ftπ ≡ σ(πs1 : s ≤ t) = FtY (= FbtY by Lemma A.5). Consequently, αt is deterministic and
πt1 f = g(t, Y (t)), with
g(t, y) = 1E (y − 1) αt f (1) + (1 − αt )f (−1) + 1O (y − 1) (1 − αt )f (1) + αt f (−1) .
Taking into account that αt0 = µ(1 − 2αt ) and that 1E (y − 1) = 1O (y), we have
∂
g(t, y) + λ y [g(t, y + 1) − g(t, y)] = (λ y + µ) (1 − 2αt ) [f (−1) − f (1)] 1E (y) − 1O (y) ,
∂t
and therefore
−1
lim h
h→0
1
E[πt+h
f
−
1
πt1 f |Ftπ ]
= (λY (t) + µ) 1E (Y (t)) − 1O (Y (t)) (1 − 2αt ) f (−1) − f (1) .
The right side is not just a function of πt1 , and it follows that π 1 is not a Markov process.
Of course, there is additional structure in the classical example with
Z t
Y (t) = σW (t) +
h(X1 (s))ds,
(3.13)
0
W a standard Brownian motion. With this example in mind, we have the following definition.
Definition 3.10 For S = S1 ×S0 with S0 = Rd and γ the projection of S onto S0 , the filtered
e each
martingale problem for (A, γ) has additive observations, if for each solution (Ye , π
e, Z),
r ≥ 0, and each y ∈ Rd ,
Yb (t) = Ye (r + t) − Ye (r) + y
b
1
|FbtY ∨ σ(e
πr1 )]
π
bt1 = E[e
πr+t
(3.14)
determines a solution (Yb , π
b1 × δYb , π
b01 × δy ) of the filtered martingale problem for (A, γ).
Lemma 3.11 Let S = S1 ×S0 with S0 = Rd and γ be the projection of S onto S0 , and suppose
that A satisfies Condition 2.1. Assume that every solution X = (X1 , Y ) of the martingale
problem for A has a version such that Y is cadlag with no fixed points of discontinuity and
for each r ≥ 0 and y ∈ Rd , (X1 (· + r), Y (· + r) − Y (r) + y) is a solution of the martingale
problem for A. Then the filtered martingale problem for (A, γ) has additive observations.
18
Remark 3.12 If X1 is the solution of the martingale problem for a generator L satisfying
Condition 2.1 with S replaced by S1 , ((aij )) = σσ > and
A[f1 f2 ] = f2 Lf1 + f1 (
1X
aij ∂i ∂j f2 + h · ∇f2 )
2 i,j
for f1 ∈ D(L) and f2 ∈ Cc2 (Rd ), that is, the S0 = Rd component satisfies (3.13) with W
independent of X1 , then the hypotheses of the lemma are satisfied.
e is a solution of the filtered martingale problem for (A, γ, µ
Proof. Suppose that (Ye , π
e, Z)
b0 ).
Then there exists a solution X = (X1 , Y ) of the martingale problem for A and a random
variable Z such that (X1 (0), Y (0), Z) has distribution µ
b0 , Y has the same distribution as
e
e and
Y , and for t ≥ 0, there exist Ht : MS0 [0, ∞) × S0 → P(S) such that π
et = Ht (Ye , Z)
Y,Z
1
πt = πt × δY (t) = Ht (Y, Z) is the conditional distribution of X(t) given Fbt .
b1 , Yb ) = (X1 (· + r), Y (· + r) − Y (r) + y) is a solution of the martingale
By assumption, (X
problems for A. Consequently, defining π
b1 by
1
g|FtY ∨ σ(πr )],
π
bt1 g = E[g(X1 (r + t))|FtY ∨ σ(πr )] = E[πr+t
b
b
(Yb , π
b1 × δYb (·) , πr ) is a solution of the filtered martingale problem for (A, γ). Since (Yb , π) has
the same distribution as (Ye (· + r) − Ye (r) + y, π
e), the filtered martingale problem for (A, γ)
has additive observations.
Theorem 3.13 Let A ⊂ B(S) × M (S) and γ : (x, y) ∈ S1 × Rd → y ∈ Rd , and assume
Condition 3.5. Suppose that the filtered martingale problem for (A, γ) has additive observae be a solution of the filtered martingale problem for (A, γ, µ
b0 ). If
tions. Let (Ye , π
e1 × δYe , Z)
1
uniqueness holds for the martingale problem for (A, ν0 ), then π
e is a P(S1 )-valued Markov
process.
Proof. As in the proof of Theorem 3.8, fix r ≥ 0, and let (Yb , π
b) be as in (3.14) with
1
1
er , they have the same distribution. By Lemma A.12, a process
y = 0. Since π
b0 = π
1∗
(Y ∗ , π 1∗ , Z ∗ ) can be constructed so that (Y ∗ (r+·)−Y ∗ (r), πr+·
, πr1∗ ) has the same distribution
1∗
on MRd ×P(S1 ) [0, ∞) × P(S1 ) as (Yb , π
b1 , π
b01 ), (Y ∗ (· ∧ r), π·∧r
, Z ∗ , πr1∗ ) has the same distribution
1
e π
on MRd ×P(S1 ) [0, r] × S0 × P(S1 ) as (Ye (· ∧ r), π
e·∧r
, Z,
er1 ), and
1∗
1∗
E[g(Y ∗ (r + ·) − Y ∗ (r), πr+·
)|Y ∗ (· ∧ r), π·∧r
, Z ∗ , πr1∗ ]
1∗
= E[g(Y ∗ (r + ·) − Y ∗ (r), πr+·
)|πr1∗ ].
Employing the assumption of additive observations, the proof that (Y ∗ , π ∗ , Z ∗ ) is a solution
of the filtered martingale problem for (A, γ, µ
b0 ) and the proof of the Markov property are
essentially the same as before.
Remark 3.14 Theorem 3.13 can be extended to processes in which S0 is a group and the
definition of Yb in (3.14) is replaced by Yb (t) = y Ye (r)−1 Ye (r + t).
19
4
Filtering equations
In the nonlinear filtering literature, a filtering equation is a collection of identities satisfied by
{πt f, f ∈ D} and the observation process Y for a set of test functions D. The set D should
be small enough to handle easily, but large enough to insure that the identities uniquely
determine π as a function of Y . Uniqueness means that if π
e is another P(S)-valued process
Y
adapted to {Ft } and {e
π f, f ∈ D} and Y satisfy the identities, then π
e = π.
The results of Section 3 can be exploited to prove uniqueness for a filtering equation
provided each solution of the filtering equation has the appropriate martingale properties.
Then, Proposition 3.1 ensures that the solution of the filtering equation satisfies
π
et = Ht (Y ) a.s.,
(4.1)
where Ht is the function that gives the conditional distribution of X(t) given FtY for some
solution of the martingale problem for A. Ht (and hence the solution of the filtering equation)
is uniquely determined provided the joint distribution of (X, Y ) is uniquely determined.
In practice, it frequently turns out that verifying that π
e “has the appropriate martingale
properties” requires a change of measure, but since the new measure is equivalent to the
original measure, (4.1) still holds. In the next section, we illustrate this argument in the
classical setting of a signal in additive white noise.
4.1
Filtering equations for a signal in additive white noise
We consider the classical Markov model with additive white noise, in which
Z t
Y (t) = Y (0) + W (t) +
h(X(s))ds,
(4.2)
0
where W is a standard d-dimensional Brownian motion and h = (h1 , · · · , hd )> is a Borel
function. Note that we are assuming S0 = Rd . Note that we are not assuming the independence of X and W .
Define
Z t
Mf (t) := f (X(t)) −
Af (X(s)) ds,
f ∈ D(A),
0
and assume that, for i = 1, · · · , d,
Z
hMf , Wi it =
t
Ci f (X(s), Y (s)) ds,
0
where Ci is an operator mapping D(A) into M (S0 × Rd ).
Remark 4.1 Note that if f = 1, then Mf (t) = 1 and therefore
hMf , Wi it = 0,
t ≥ 0,
that is, Ci 1 = 0.
In the uncorrelated case, that is, when X and W are independent, Ci f = 0, for all
f ∈ D(A).
20
Example 4.2 Let X be a diffusion process with values in Rm solving
dX(t) = b(X(t)) dt + σ(X(t)) dW (t) + σ(X(t)) dW (t),
(4.3)
with W a Wiener process, independent of W . Then
Ci f (x, y) =
m
X
∂j f (x)σj,i (x),
f ∈ Cc2 (Rm ),
j=1
where ∂j f denotes the partial derivative of f with respect to the j-th component of x.
Setting Cf = (C1 f, · · · , Cp f ), the pair (X, Y ) will be a solution of the martingale problem
b given by
(or local martingale problem) for A
1
∆ϕ(y) + h(x) · ∇ϕ(y) + Cf (x, y)∇ϕ(y),
Af ⊗ ϕ(x, y) = Af (x) ϕ(y) + f (x)
2
with domain
b = {f ⊗ ϕ : f ∈ D(A), ϕ ∈ C ∞ (Rd )}.
D(A)
c
Since we need the joint distribution of X and Y to be determined, we need uniqueness for
b and with Condition 2.1(iii) in mind, we assume
the martingale problem for A,
Condition 4.3 There exist a function ψ0 (x, y) ≥ 1 and constants cf such that for each
i = 1, · · · , d and f ∈ D(A),
|Ci f (x, y)| ≤ cf ψ0 (x, y).
(4.4)
Let ν0 denote the distribution for X(0) and νb0 the distribution for (X(0), Y (0)).
Condition 4.4 The process X is a cadlag process with values in S and is a solution of the
martingale problem for (A, ν0 ), where the operator A satisfies Condition 2.1 and uniqueness
b νb0 ).
holds for the stopped martingale problem for (A,
b satisfies Condition 2.1 with ψ replaced by
Remark 4.5 Note that under these conditions, A
b y) = ψ(x) + ψ0 (x, y) + |h(x)|.
ψ(x,
b is implied by uniqueness for A by a slight
In the uncorrelated case, uniqueness for A
modification of Lemma 4.4 of Kurtz and Ocone (1988). In particular, if (X, Y ) is a solution
b then for
of the (local) martingale problem for A,
Z t
Z(t) = Y (t) − Y (0) −
h(X(s))ds,
0
(X, Z) is a solution of the (local) martingale problem for (X, W ), that is, for the operator
b0 obtained by setting h = 0 in the definition of A.
b Consequently, uniqueness for A
b0 implies
A
0
b In the uncorrelated case, uniqueness for A
b follows by Theorem 4.10.1 of
uniqueness for A.
Ethier and Kurtz (1986).
b can be found in Theorem
In the correlated case, a sufficient condition for uniqueness for A
4.2 of Bhatt and Karandikar (1999) under the assumption that for each i and f ∈ D(A),
Ci f (x, y) is a function of x alone. In the diffusion case, Example 4.2, Cherny (2003) has
shown joint uniqueness in law for the solution (X, W, W ) of (4.3) which in turn implies
b0 and hence uniqueness for A.
b
uniqueness for the martingale problem for A
21
Finally, we assume the following integrability conditions.
Condition 4.6 For each t > 0,
Z t
(ψ(X(s)) + ψ0 (X(s), Y (s)) + |h(X(s))|) ds < ∞,
E
t ≥ 0,
(4.5)
0
and
Z
t
(πs ψ + (πs ψ0 (·, Y (s)))2 + (πs |h|)2 )ds < ∞
a.s.
(4.6)
0
Rt
Under Condition 4.6, the innovation processhI π (t) = Y (t)−Yi(0)− 0 πshh ds is a Brownian
i
Rt
Rt
motion. (Note that we are not assuming that E 0 |h(X(s))|2 ds < ∞. E 0 |h(X(s))|ds <
∞ is sufficient to ensure that I π is a martingale.) For all f ∈ D(A), π satisfies
Z t
πt f = π0 f +
πs Af ds
0
Z t
[πs (hf + Cf (·, Y (s)) − πs h πs f ] [dY (s) − πs h ds] ,
+
0
where Cf (x1 , y) = (C1 f (x1 , y), · · · , Cp f (x1 , y)), or equivalently,
Z t
Z t
πt f = π 0 f +
πs Af ds +
[πs (hf + Cf (·, Y (s)) − πs h πs f ] dI π (s).
0
(4.7)
0
Define the unnormalized filter for X as the M(S)-valued process
ρt (dx) := Ztπ πt (dx),
where
Ztπ
Z
:= exp
0
t
1
πs (h) dY (s) −
2
Z
t
(4.8)
|πs (h)| ds .
2
(4.9)
0
By Itô’s formula, one can show that ρ satisfies the Duncan-Mortensen-Zakai unnormalized
filtering equation,
Z t
Z t
ρ t f = π0 f +
ρs Af ds +
ρs hf + Cf (·, Y (s)) dY (s), ∀f ∈ D(A).
0
0
The following result, essentially Theorem 9.1 of Bhatt, Kallianpur, and Karandikar
(1995), extends Theorems 4.1 and 4.5 of Kurtz and Ocone (1988) and gives uniqueness
of the Kushner-Stratonovich and Fujisaki-Kallianpur-Kunita equations.
Theorem 4.7 Assume that Conditions 4.3, 4.4, and 4.6, are satisfied. Let {µt } be a {FtY }adapted, cadlag P(S)-valued process satisfying
Z t
(µs ψ + (µs ψ0 (·, Y (s)))2 + (µs |h|)2 )ds < ∞, a.s.,
(4.10)
0
22
and for f ∈ D(A),
t
Z
µ t f = π0 f +
µs Af ds
0
Z
+
t
µs hf + Cf (·, Y (s)) − µs hµs f dI µ (s),
(4.11)
0
where I µ (t) = Y (t) −
Rt
0
µs h ds. Then µt = πt , t ≥ 0, a.s.
Remark 4.8 There is a large literature on uniqueness for the filtering equations with varying
assumptions depending on the techniques used by the authors. Kurtz and Ocone (1988),
Bhatt, Kallianpur, and Karandikar (1995), Lucic and Heunis (2001), and Rozovskiı̆ (1991)
provide a reasonable sampling of results and methods. Our introduction and exploitation of
the filtered local-martingale problem allows us to avoid a number of assumptions that appear
in many of the earlier results. In particular,
R t we do not assume that h is continuous. We only
require the first moment assumption E[ 0 |h(X(s))|ds] < ∞ rather than a second moment
assumption. (Note that there is no expectation in (4.6) and (4.10).) There are no a priori
moment assumptions on the solution µ (only on the true conditional distribution).
Proof. If µ is a solution of (4.11), then µ
b = µ × δY satisfies
Z t
b ⊗ ϕ ds
µ
bs Af
µ
bt f ⊗ ϕ = ϕ(Y (0))π0 f +
0
Z th
ϕ(Y (s))µs hf + Cf (·, Y (s))
+
0
−ϕ(Y (s)µs hµs f C + µs f ∇ϕ(Y (s))T dI µ (s).
If I µ is a {FtY }-local martingale, then
Z
µ
bt f ⊗ ϕ − ϕ(Y (0))π0 f −
0
t
b ⊗ ϕ ds
µ
bs Af
is a {FtY }-local martingale, and Proposition 3.1 implies that there exist a solution (X ∗ , Y ∗ )
b νb0 ) and a P(S)-valued process π ∗ such that (π ∗ , Y ∗ )
of the local martingale problem for (A,
and (µ, Y ) have the same distribution and πt∗ is the conditional distribution of X ∗ (t) given
∗
FtY . In addition, there exists Ht such that πt∗ = Ht (Y ∗ ) and µt = Ht (Y ). Since uniqueness
b νb0 ), (X ∗ , Y ∗ ) has the same distribution as
holds for the local martingale problem for (A,
(X, Y ) and µt = Ht (Y ) = πt .
Unfortunately, it is Rnot immediately clear that I µ is a local martingale. However, since
t
I π (t) = Y (t) − Y (0) − 0 πs hds is a Brownian motion, if we define
ξ(t) = µt h − πt h,
Z t
τn = inf{t :
(|ξ(s)|2 + |µs h|2 )ds ≥ n},
0
23
and let Qn be the probability measure on Yn∧τn given by
Z n∧τn
Z
dQn
1 n∧τn
T
π
ξ(s) dI (s) −
= exp{
|ξ(s)|2 ds},
dP
2 0
0
then under Qn ,
µ
Z
π
t∧τn
I (t ∧ τn ) = I (t ∧ τn ) −
ξ(s)ds,
0
is a martingale for 0 ≤ t ≤ n. Consequently, under Qn ,
Z t∧τn
b ⊗ ϕ ds
µ
bs Af
µ
bt∧τn f ⊗ ϕ − ϕ(Y (0))π0 f −
(4.12)
0
is a {FtY }-martingale, and by Proposition 3.4 and uniqueness of the stopped martingale
b
problem for A,
µt 1{n∧τn ≥t} = Ht (Y )1{n∧τn ≥t} = πt 1{n∧τn ≥t} .
Y
It follows that Qn = P on Fn∧τ
, n = 1, 2, . . . and hence by (4.10), τn → ∞ a.s. and µt = πt
n
a.s.
Corollary 4.9 Assume that Conditions 4.3, 4.4, and 4.6 are satisfied. Let {θt } be a {FtY }adapted, cadlag M(S)-valued process satisfying
Z t
(θs ψ + (θs ψ0 (·, Y (s)))2 + (θs |h|)2 )ds < ∞, a.s.,
0
such that for every f ∈ D(A),
Z t
Z t
θt f = π0 f +
θs Af ds +
θs hf + Cf (·, Y (s)) dY (s).
0
(4.13)
0
Then θt = ρt a.s. for all t ≥ 0.
Proof. For > 0, define β = inf{t > 0 : θt 1 ≤ } and set
θt
and 0 ≤ t < β0 ≡ lim β .
→0
θt 1
Then, by Itô’s formula, µ satisfies (4.11) on [0, β0 ). Defining τn and the appropriate change
of measure as in the proof of the previous theorem, it follows that under the new measure,
(4.12), with τn replaced by τn ∧ β is a martingale, and as before, µt 1{τn ∧β ≥t} = πt 1{τn ∧β ≥t} .
Letting n → ∞, µt∧β = πt∧β .
Observe that, for t ≤ β ,
Z t
θt 1 = 1 +
θs 1 πs hdY (s),
µt =
0
so
Z
1 t
θt 1 =
= exp{ πs hdY (s) −
|πs h|2 ds},
2
0
0
Since for T > 0, inf t≤T Ztπ > 0, it follows that β0 = ∞ and hence
Ztπ
Z
t
t < β0 .
µt = πt and θt = θt 1 πt = ρt .
24
4.2
Related results on filtering equations
The filtered martingale problem was first introduced in Kurtz and Ocone (1988) in the
special case considered here, following
a question raised by Giorgio Koch: For all f in the
Rt
domain of A, the process πt f − 0 πs Af ds is a {FtY }-martingale; can this observation be
used to study nonlinear filtering problems? Under the condition that the state space S is
locally compact, the results on the filtered martingale problem in Kurtz and Ocone (1988)
give uniqueness of Zakai and Kushner-Stratonovich equations in the natural class of {FtY }adapted measure-valued processes. Stochastic equations relate known random inputs (in
our case Y ) to unknown random outputs (in our case, the conditional distribution π). Weak
uniqueness (or more precisely, joint uniqueness in law) says that the joint distribution of
the input and the output is uniquely determined by the distribution of the input. Strong or
pathwise uniqueness says that there is a unique, appropriately measurable transformation
that maps the input into the output. In our case, since the equations are derived to be
satisfied by the conditional distribution, existence of a transformation (Ht of Proposition 3.1
and Theorem 3.6) is immediate. Consequently, it follows by a generalization of a theorem
of Engelbert (1991) (see Kurtz (2007), Theorem 3.14) that weak and strong uniqueness are
equivalent.
The uniqueness results derived using the filtered martingale problem in Kurtz and Ocone
(1988) were extended in Bhatt, Kallianpur, and Karandikar (1995), in particular, eliminating
the local compactness assumption on the state space of the signal. Still in the framework
of signals observed in Gaussian white noise, Bhatt and Karandikar (1999) goes beyond the
classical Markov model with additive white noise and considers diffusive, non-Markovian
signal/observation systems. The systems solve stochastic differential equations with coefficients that may depend on the signal and on the whole past trajectory of the observation
process. Therefore, in particular, the signal need not be a Markov process. By enlarging the
observation space to a suitable space of continuous functions, such a system can be seen as
the solution of a martingale problem, and the filtered martingale problem approach can be
used.
In Kallianpur and Mandal (2002) the signal is the solution of a stochastic delay-differential
equation, and the Wiener processes driving the signal and the observations are independent.
In this case the signal state space is enlarged to a suitable space of continuous functions, and
the system is seen as the solution of a martingale problem, and again the filtered martingale
problem approach can be used.
Filtering models with point process observations can also be analyzed using the filtered martingale problem approach. In Kliemann, Koch, and Marchetti (1990), the signal/observation system is a Markov process, and the signal is a jump-diffusion process (not
necessarily Markovian by itself), while the observation process is a counting process, with
unbounded intensity. Strong uniqueness for the filtering equation is obtained in a class
of probability-valued processes characterized by a suitable second moment growth condition. In Ceci and Gerardi (2001a), the observation is still a counting process, while the
signal/observation system is a Markov jump process with values in S = Rd × N (see also
Ceci and Gerardi (2000), where the signal process itself is a jump Markov process); the weak
uniqueness of the Kushner-Stratonovich equation is obtained by the filtered martingale problem approach, while the pathwise uniqueness is obtained by a direct method. (Note that
25
the notion of weak solution considered in this paper places additional restrictions on the
solution beyond adaptedness and satisfying the identity. Consequently, the equivalence of
weak and strong solutions mentioned above does not hold in this context.) Uniqueness of
the filtered martingale problem has also been used for a partially observable control problem
of a jump Markov system with counting observations in Ceci, Gerardi, and Tardelli (2002)
(see also Ceci and Gerardi (1998) and Ceci and Gerardi (2001b)). The case of a marked
point observation process has been considered in various papers. In Fan (1996), the signal/observation system is a continuous-time Markov chain, with S a finite set. In Ceci and
Gerardi (2005) and Ceci and Gerardi (2006a), partially observed branching processes are
discussed, while Ceci and Gerardi (2006b) focuses on the financial applications of filtering.
In all these examples, the observation state space S0 is discrete but not necessarily finite.
5
Constrained Markov processes
In this section, M(S × [0, ∞)) will denote the space of Borel measures µ on S × [0, ∞) such
that µ(S × [0, t]) < ∞ for each t > 0.
Let A and B be operators satisfying Condition 2.1 with ψ replaced by ψA and ψB ,
respectively, and D(A) = D(B) = D, and let D and ∂D be closed subsets of S. In many
situations, ∂D will be the topological boundary of D, but that is not necessary.
Definition 5.1 A measurable, D-valued process X and a random measure Γ in M(∂D ×
[0, ∞)) give a solution of the constrained martingale problem for (A, B, D, ∂D) if there exists
a filtration {Ft } such that X and Γ are {Ft }-adapted,
Z
Z t
ψB (x)Γ(dx × ds)] < ∞, t ≥ 0,
E[ ψA (X(s))ds +
∂D×[0,t]
0
and for each f ∈ D,
Z
t
Z
Bf (x)Γ(dx × ds)
Af (X(s))ds −
f (X(t)) −
0
∂D×[0,t]
is an {Ft }-martingale. For ν0 ∈ P(D), (X, Γ) is a solution of the constrained martingale
problem for (A, B, D, ∂D, ν0 ) if (X, Γ) is a solution of the constrained martingale problem
for (A, B, D, ∂D) and X(0) has distribution ν0 .
A P(D)-valued
function {νt } is a solution of the forward equation for (A,RB, D, ∂D) if for
Rt
each t > 0, 0 νs ψA ds < ∞ and there exists µ ∈ M(∂D × [0, ∞)) such that ∂D×[0,t] ψB dµ <
∞, t > 0, and for each f ∈ D,
Z t
Z
νt f = ν0 f +
νs Af ds +
Bf (x)µ(dx × ds), t ≥ 0.
0
∂D×[0,t]
Remark 5.2 By uniqueness for a constrained martingale problem, we mean uniqueness of
the finite-dimensional distributions of X. We do not expect Γ to be unique. For example,
26
there may be a measure µ
b such that
constrained martingale problem and
R
∂D
Bf db
µ ≡ 0. Then, if (X, Γ) is a solution of the
b
Γ(dx
× ds) = Γ(dx × ds) + µ
b(dx)ds,
b is also a solution.
then (X, Γ)
The most familiar examples of constrained Markov processes are reflecting diffusion processes satisfying equations of the form
Z t
Z t
Z t
X(t) = X(0) +
σ(X(s))dW (s) +
b(X(s))ds +
m(X(s))dξ(s),
0
0
0
where X is required to remain in the closure of a domain D ⊂ Rd and ξ is a nondecreasing
process that increases only when X is on the topological boundary ∂D of D. Then
∂2
1X
aij (x)
f (x) + b(x) · ∇f (x),
2 i,j
∂xi ∂xj
Af (x) =
where a(x) = ((aij (x))) = σ(x)σ(x)T ,
Bf (x) = m(x) · ∇f (x),
and
Z
t
Γ(C × [0, t]) =
1C (X(s))dξ(s).
0
As before, let γ : S → S0 , Y (t) = γ(X(t)), and
πt (Γ) = P{X(t) ∈ Γ|FbtY,Z }.
Then
t
Z
Z
b
Bf (x)Γ(dx
× ds)
πs Af ds −
πt f −
0
∂D×[0,t]
b is the dual predictable projection of Γ with respect to
is a {FbtY,Z }-martingale, where Γ
Y,Z
{Fbt }. (See Kurtz and Stockbridge (2001), Lemma 6.1.)
Definition 5.3 Let µ
b0 ∈ P(D × S0 ).
e π
e ∈ MS0 [0, ∞) × M(∂D × [0, ∞)) × MP(D) [0, ∞) × S0
(Ye , Γ,
e, Z)
is a solution of the filtered martingale problem for (A, B, D, ∂D, γ, µ
b0 ), if
e =µ
E[e
π0 (C)1D (Z)]
b0 (C × D),
C ∈ B(D), D ∈ B(S0 ),
e
e are {FbtYe ,Ze }-adapted, for each g ∈ B(D) and t ≥ 0,
π
e0 is σ(Z)-measurable,
π
e and Γ
Z t
Z t
π
es (g ◦ γ)ds =
g(Ye (s))ds ,
0
0
27
(5.1)
Z t
Z
E[ π
es ψA ds +
0
e
ψB (x)Γ(dx
× ds)] < ∞,
t ≥ 0,
∂D×[0,t]
and for each f ∈ D,
Z
π
et f −
0
t
Z
π
es Af ds −
e
Bf (x)Γ(dx
× ds)
∂D×[0,t]
is an {FbtY ,Z }-martingale.
e π
e satisfies all the conditions except (5.1), we will refer to it as a solution of
If (Ye , Γ,
e, Z)
the filtered martingale problems for (A, B, D, ∂D, γ).
e e
The following extension of Lemma 2.8 follows from Corollary 1.12 of Kurtz and Stockbridge (2001).
Lemma 5.4 Let A and B satisfy Condition 2.1 with ψ replaced by ψA and ψB respectively.
If {νt } is a solution of the forward equation for (A, B, D, ∂D), then there exists a solution
(X, Γ) of the constrained martingale problem for (A, B, D, ∂D) such that νt f = E[f (X(t))].
b as in Section 3, and define B
b by
Define A
b f0
B[f
m
Y
gki ](x, z, u) =
f0 (z)
m
Y
!
gki (uki ) Bf (x) .
k,i=1
k,i=1
b and B.
b Any solution (Ye , Γ,
e π
e of
As before, if A and B satisfy Condition 2.1, then so do A
e, Z)
the filtered martingale problem for (A, B, D, ∂D, γ, µ
b0 ) determines a solution of the forward
∞
∞
b
b
equation for (A, B, D × S0 × [0, 1] , ∂D × [0, 1] , νb0 ), which, by Lemma 5.4 corresponds to a
solution of the constrained martingale problem satisfying (3.4). This observation then gives
the following analog of Theorem 3.6.
Theorem 5.5 Let A and B satisfy Condition 2.1, µ
b0 ∈ P(S × S0 ), and γ : S → S0
e
e
e
be Borel measurable. Let (Y , Γ, π
e, Z) be a solution of the filtered martingale problem for
(A, B, D, ∂D, γ, µ
b0 ). Then the following hold:
a) There exists a solution (X, Γ) of the constrained martingale problem for (A, B, D, ∂D)
and an S0 -valued random variable Z such that (X(0), Z) has distribution µ
b0 and Ye has
the same distribution on MS0 [0, ∞) as Y = γ ◦ X.
b) For each t ≥ 0, there exists a Borel measurable mapping Ht : MS0 [0, ∞) × S0 → P(S)
such that πt = Ht (Y, Z) is the conditional distribution of X(t) given FbtY,Z , and π
et =
e
e
Ht (Y , Z) a.s. In particular, π
e has the same finite-dimensional distributions as π.
c) If Y and Ye have sample paths in DS0 [0, ∞), then Y and Ye have the same distribution
on DS0 [0, ∞) and Ht is Borel measurable mapping from DS0 [0, ∞) × S0 to P(S).
28
d) If uniqueness holds for the constrained martingale problem for (A, B, D, ∂D, ν0 ), then
uniqueness holds for the filtered martingale problem for (A, B, D, ∂D, γ, µ
b0 ) in the sense
e
e
e
that if (Y, Γ, π, Z) and (Y , Γ, π
e, Z) are solutions, then for each 0 ≤ t1 < · · · < tm ,
e have the same distribution on P(S)m ×
(πt1 , . . . , πtm , Y, Z) and (e
πt1 , . . . , π
etm , Ye , Z)
MS0 [0, ∞) × S0 .
The analogs of Theorems 3.8 and 3.13 hold by essentially the same arguments as before.
A
A.1
Appendix
A martingale lemma
Let {Ft } and {Gt } be filtrations with Gt ⊂ Ft .
Lemma A.1 Suppose U and V are measurable and {Ft }-adapted, E[|U (t)| +
∞, t ≥ 0, and
Z t
U (t) −
V (s)ds
Rt
0
|V (s)|ds] <
0
is an {Ft }-martingale. Then
t
Z
E[U (t)|Gt ] −
E[V (s)|Gs ]ds
0
is a {Gt }-martingale, where we take E[V (s)|Gs ] to be the optional projection of V .
Proof. The lemma follows by the definition and properties of conditional expectations. Example A.2 If X is a solution (wrt {Ft }) of the martingale problem for A and
πt (Γ) = P{X(t) ∈ Γ|Gt },
then
Z
πt f −
t
πs Af ds
0
is a {Gt }-martingale.
A.2
Filtrations generated by processes
Let Y be a measurable stochastic process. FtY will denote the completion of σ(Y (s), s ≤ t)
and FbtY will denote the completion of
Z s
σ(
h(Y (r))dr, s ≤ t, h ∈ B(S)) ∨ σ(Y (0)).
0
Lemma A.3 If Y is {FtY }−progressively measurable, then FbtY ⊂ FtY .
29
Lemma A.4 For almost every t, Y (t) is FbtY .
Proof. For g ∈ B(S0 ),
Z
t
Z
g(Y (s))ds −
Mg (t) =
0
t
E[g(Y (s))|FbsY ]ds
0
is a continuous {FbtY }-martingale. (Take E[g(Y (s))|FbsY ] to be the optional projection of
g ◦ Y .) Since Mg is a finite variation process, it must be zero with probability one, and
hence, with probability one
g(Y (t)) = E[g(Y (t))|FbtY ] for almost every t,
which in turn implies that for almost every t, g(Y (t)) = E[g(Y (t))|FbtY ] a.s. and g(Y (t))
is FbtY -measurable. Since S0 is separable, there exists a countable separating set {gk } such
that for almost every t, g1 (Y (t)), g2 (Y (t)), . . . are FbtY -measurable and hence Y (t) is FbtY measurable.
Lemma A.5 If Y is cadlag with no fixed points of discontinuity (that is, P{Y (t) = Y (t−)} =
1 for all t), then FtY = FbtY , t ≥ 0.
Proof. If Y is cadlag, it is {FtY }-progressively measurable so FbtY ⊂ FtY . Y (0) is Fb0Y
measurable by definition. For t > 0, since P{Y (t) = Y (t−)} = 1, for g ∈ Cb (S),
Z
1 t
g(Y (t)) = lim
g(Y (r))dr a.s.,
→ t−
and hence Y (t) is FbtY -measurable. Consequently, FtY ⊂ FbtY .
The following lemma implies that most cadlag processes of interest will have no fixed
points of discontinuity.
Lemma A.6 Let U and V be S-valued random variables and let G be a σ-algebra of events.
Suppose that M ⊂ Cb (S) is separating and
E[f (U )|G] = f (V )
(A.1)
for all f ∈ M . Then U = V a.s.
In particular, if Y is cadlag and adapted to {Gt } and
lim E[f (Y (t))|Gs ] = E[f (Y (t))| ∨s<t Gs ] = f (Y (t−)),
s→t−
for f ∈ M , then Y (t) = Y (t−) a.s.
Remark A.7 Note that if
Z
f (Y (t)) −
t
Zf (s)ds
0
is a martingale for Zf satisfying E[
Rt
0
|Zf (s)|ds] < ∞, then (A.2) holds.
30
(A.2)
Proof. Let Z be a nonnegative G-measurable random variable. Then E[f (U )Z] = E[f (V )Z]
for f ∈ M and since M is separating, this identity must hold for all f ∈ B(S). Consequently,
(A.1) holds for all f ∈ B(S), and replacing f by f 2 ,
E[f 2 (U )] = E[f 2 (V )] = E[E[f (U )|G]f (V )] = E[f (U )f (V )]
and hence that E[(f (U ) − f (V ))2 ] = 0 for all f ∈ B(S).
A.3
Stopped filtrations and filtrations generated by stopped processes
Let V be a cadlag process, FtV be the completion of σ(V (s), s ≤ t), and S be the collection
of finite, {FtV }-stopping times. For τ ∈ S, define GτV to be the completion of σ(V (t ∧ τ ) :
t ≥ 0) ∨ σ(τ ). Of course, GtV = FtV , and more generally GτV = FτV for all discrete stopping
times in S and GτV ⊂ FτV for all τ ∈ S, but we do not know whether or not GτV = FτV for all
τ ∈ S.
Lemma A.8 If τ1 , τ2 ∈ S, then
FτV1 ∧τ2 = {(A1 ∩ {τ1 < τ2 }) ∪ (A2 ∩ {τ1 ≥ τ2 }) : A1 ∈ FτV1 , A2 ∈ FτV2 }
(A.3)
and for τ ∈ S and t ≥ 0,
GτV∧t = {(A1 ∩ {τ < t}) ∪ (A2 ∩ {τ ≥ t}) : A1 ∈ GτV , A2 ∈ GtV = FtV }.
(A.4)
Remark A.9 The first assertion holds for arbitrary filtrations.
Proof. Since FτV1 ∧τ2 ⊂ FτV1 ∩ FτV2 , is follows that FτV1 ∧τ2 is contained in the right side of (A.3).
(Take A1 = A2 ∈ FτV1 ∧τ2 .)
Observe that {τ1 < τ2 } ∈ FτV1 ∧τ2 since
{τ1 < τ2 } ∩ {τ1 ∧ τ2 ≤ t} = {τ1 ≤ t < τ2 } ∪ ∪r∈Q,r≤t {τ1 ≤ r < τ2 } ∈ FtV .
Now let A1 ∈ FτV1 . Then
A1 ∩ {τ1 < τ2 } ∩ {τ1 ∧ τ2 ≤ t} = A1 ∩ {τ1 < τ2 } ∩ {τ1 ≤ t} ∈ FtV ,
so A1 ∩ {τ1 < τ2 } ∈ FτV1 ∧τ2 , and for A2 ∈ FτV2 ,
A2 ∩ {τ1 ≥ τ2 } ∩ {τ1 ∧ τ2 ≤ t} = A2 ∩ {τ1 ≥ τ2 } ∩ {τ2 ≤ t} ∈ FtV ,
so A2 ∩ {τ1 ≥ τ2 } ∈ FτV1 ∧τ2 .
If A1 ∈ GτV , then A1 differs from a set of the form {(V (·∧τ ), τ ) ∈ C}, C ∈ B(DE [0, ∞))×
B([0, ∞)), by an event of probability zero, and, noting that {τ ∧ t < t} = {τ < t},
{(V (· ∧ τ ), τ ) ∈ C} ∩ {τ < t} = {(V (· ∧ τ ∧ t), τ ∧ t) ∈ C} ∩ {τ < t} ∈ GτV∧t .
31
Similarly,
{V (· ∧ t) ∈ C} ∩ {τ ≥ t} = {V (· ∧ τ ∧ t) ∈ C} ∩ {τ ≥ t} ∈ GτV∧t ,
and the right side of (A.4) is contained in the left.
Finally,
{(V (· ∧ τ ∧ t), τ ∧ t) ∈ C} ∩ {τ < t} = {(V (· ∧ τ ), τ ) ∈ C} ∩ {τ < t} ∈ GτV
and
{(V (· ∧ τ ∧ t), τ ∧ t) ∈ C} ∩ {τ ≥ t} = {(V (· ∧ t), t) ∈ C} ∩ {τ ≥ t} ∈ GtV ,
so the left side of (A.4) is contained in the right.
We have the following consequence of the previous lemma.
Lemma A.10 Let E[|Z|] < ∞ and τ ∈ S. Then
E[Z|FτV∧t ] = E[Z|FτV ]1{τ <t} + E[Z|FtV ]1{τ ≥t}
and
E[Z|GτV∧t ] = E[Z|GτV ]1{τ <t} + E[Z|GtV ]1{τ ≥t} ,
and since GtV = FtV ,
A.4
E[Z|FτV∧t ]1{τ ≥t} = E[Z|GτV∧t ]1{τ ≥t} .
Random probability measures as conditional distributions
Lemma A.11 Let π
e be a P(S)-valued random variable and Ze a S0 -valued random variable
e F,
e Pe). Then there exists a probability space with random variables
on a probability space (Ω,
(X, Z, π) in S × S0 × P(S), and a sub-σ-algebra D such that (Z, π) has the same distribution
e π
as (Z,
e), Z is D-measurable, and π is the conditional distribution of X given D.
Proof. For C ∈ B(S) and D ∈ B(S0 × P(S)),
e π
ν(C, D) = E[e
π (C)1D (Z,
e)]
defines a bimeasure on S × (S0 × P(S)). By Morando’s theorem (see, for example, Ethier
and Kurtz (1986), Appendix 8), ν extends to a probability measure on S × S0 × P(S). Let
(X, Z, π) be the coordinate random variables on (S × S0 × P(S), B(S × S0 × P(S)), ν). Then
e π
by definition, (Z, π) has the same distribution as (Z,
e), and defining D = σ(Z, π), the fact
that
e π
E[1C (X)1D (Z, π)] = E[e
π (C)1D (Z,
e)] = E[π(C)1D (Z, π)]
implies that π is the conditional distribution of X given D.
32
A.5
A coupling lemma
Lemma A.12 Let S0 , S1 , and S2 be complete, separable metric spaces, and let µ ∈ P(S0 , S1 )
and ν ∈ P(S0 , S2 ) satisfy µ(· × S1 ) = ν(· × S2 ). Then there exists a probability space and
random variables X0 , X1 , X2 such that (X0 , X1 ) has distribution µ, (X0 , X2 ) has distribution
ν, and
E[g(X2 )|X0 , X1 ] = E[g(X2 )|X0 ], g ∈ B(S2 ),
which is equivalent to
E[g1 (X1 )g2 (X2 )|X0 ] = E[g1 (X1 )|X0 ]E[g2 (X2 )|X0 ],
g1 ∈ B(S1 ), g2 ∈ B(S2 ).
Proof. The lemma is a consequence of Ethier and Kurtz (1986), Lemma 4.5.15
A.6
The Markov property
Lemma A.13 Let A ⊂ B(S) × M (S) and ν0 ∈ P(S). Suppose that there exists ψ ≥ 1 such
that for each f ∈ D(A), there exists a constant af such that |Af (x)| ≤ af ψ(x). Assume
that uniqueness holds for the local-martingale problem for (A, ν0 ). If X is a solution of the
local-martingale problem for (A, ν0 ) with respect to the filtration
Z s
Ft = σ(X(s) : s ≤ t) ∨ σ(
h(X(r))dr : s ≤ t, h ∈ B(S)),
0
then X is an {Ft }-Markov process.
Rt
Proof. Let Y (t) = 0 ψ(X(s))ds, and for f ∈ D(A) and g ∈ Cc1 [0, ∞), define
b g)(x, y) = g(y)Af (x) + ψ(x)f (x)g 0 (y).
A(f
b g)(x, y)| ≤
Note that since g has compact support, there exists a constant af,g such that |A(f
−2
af,g (1 + y) ψ(x), and
Z t
(1 + Y (s))−2 ψ(X(s))ds ≤ 1.
0
Consequently, if X is the unique solution of the local-martingale problem for (A, ν0 ), then
b ν0 ×δ0 ). The proof of Theorem
(X, Y ) is the unique solution of the martingale problem for (A,
A.5 in Kurtz (1998) remains valid after replacing the assumption A ⊂ B(S) × B(S) with
A ⊂ B(S) × M (S),
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